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On the implosion of a compressible fluid. I: Smooth self-similar inviscid profiles. (English) Zbl 1497.35384

Summary: In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension \(d\geq 2\) with decaying density at spatial infinity. The phase portrait of the nonlinear ODE governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated acoustic cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic \(\mathcal{C}^\infty\) self-similar solutions with suitable decay at infinity. The \(\mathcal{C}^\infty\) regularity is used in a fundamental way in our companion paper [ibid. 196, No. 2, 779–889 (2022; Zbl 1497.35385)] in the analysis of the associated linearized operator and leads, in turn, to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions \(d=2,3\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q31 Euler equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35C06 Self-similar solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
76Q05 Hydro- and aero-acoustics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Citations:

Zbl 1497.35385
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References:

[1] Bizo\'{n}, Piotr; Maison, Dieter; Wasserman, Arthur, Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity. Nonlinearity, 20, 2061-2074 (2007) · Zbl 1130.35093 · doi:10.1088/0951-7715/20/9/003
[2] Collot, Charles; Rapha\"{e}l, Pierre; Szeftel, Jeremie, On the stability of type {I} blow up for the energy super critical heat equation, Mem. Amer. Math. Soc.. Memoirs of the Amer. Math. Soc., 260, v+97 pp. (2019) · Zbl 1430.35138 · doi:10.1090/memo/1255
[3] Courant, R.; Friedrichs, K. O., Supersonic {F}low and {S}hock {W}aves, xvi+464 pp. (1948) · Zbl 0041.11302
[4] Guderley, G., Starke kugelige und zylindrische {V}erdichtungsst\"{o}sse in der {N}\"{a}he des {K}ugelmittelpunktes bzw. der {Z}ylinderachse, Luftfahrtforschung. Luftfahrtforschung, 19, 302-311 (1942) · Zbl 0061.45804
[5] Krieger, Joachim; Schlag, Wilhelm, Large global solutions for energy supercritical nonlinear wave equations on {\( \Bbb R^{3+1}\)}, J. Anal. Math.. Journal d’Analyse Math\'{e}matique, 133, 91-131 (2017) · Zbl 1402.35180 · doi:10.1007/s11854-017-0029-0
[6] Lepin, L. A., Self-similar solutions of a semilinear heat equation, Mat. Model.. Matematicheskoe Modelirovanie, 2, 63-74 (1990) · Zbl 0972.35506
[7] Malgrange, Bernard, Sur le th\'{e}or\`eme de {M}aillet, Asymptotic Anal.. Asymptotic Analysis, 2, 1-4 (1989) · Zbl 0693.34004 · doi:10.3233/ASY-1989-2101
[8] Merle, Frank; Rapha\"{e}l, Pierre; Rodnianski, Igor, On the implosion of a three dimensional compressible fluid {II: S}ingularity formation, Ann. of Math. (2). Annals of Mathematics, 196, 779-889 (2022) · Zbl 1497.35385 · doi:10.4007/annals.2022.196.2.4
[9] Merle, Frank; Rapha\"{e}l, Pierre; Rodnianski, Igor, On blow up for the energy super critical defocusing nonlinear {S}chr{\"{o}}dinger equations, Invent. Math.. Inventiones Mathematicae, 227, 247-413 (2022) · Zbl 1487.35353 · doi:10.1007/s00222-021-01067-9
[10] Meyer-ter-Vehn, J.; Schalk, C., Self-similar spherical compression waves in gas dynamics, Z. Naturforsch. A. Zeitschrift f\"{u}r Naturforschung. A, 37, 955-969 (1982) · Zbl 0583.76083
[11] Ramis, J.-P., D\'{e}vissage {G}evrey. Journ\'{e}es {S}inguli\`eres de {D}ijon, Ast\'{e}risque, 59, 4-173 (1978) · Zbl 0409.34018
[12] Sedov, L. I., Similarity and Dimensional Methods in Mechanics, 424 pp. (1982) · Zbl 0526.76002
[13] translated from the Japanese by the author, Linear differential equations in the complex domain: problems of analytic continuation, Transl. Math. Monogr., 82, xiv+269 pp. (1990) · Zbl 1145.34378 · doi:10.1090/mmono/082
[14] Troy, William C., The existence of bounded solutions of a semilinear heat equation, SIAM J. Math. Anal.. SIAM Journal on Mathematical Analysis, 18, 332-336 (1987) · Zbl 0655.35039 · doi:10.1137/0518026
[15] Zel \('\) dovich, Ya. B.; Razier, Yu. P., Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, Volume II (1967)
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